Congruent triangles
Congruent figures have exactly the same size and shape. For triangles, a few matching parts are enough to guarantee all the rest match, the basis of many proofs.
Congruent figures
Two figures are congruent if one can be moved onto the other by slides, flips, and turns. Corresponding parts of congruent figures are congruent. In a congruence statement such as triangle ABC is congruent to triangle DEF, the order of letters shows the correspondence: A matches D, B matches E, and C matches F.
SSS
Side-side-side (SSS): if three sides of one triangle are congruent to three sides of another, the triangles are congruent. This is why triangles are rigid; three fixed side lengths allow only one shape.
SAS
Side-angle-side (SAS): if two sides and the included angle, the angle between them, of one triangle are congruent to those of another, the triangles are congruent. The angle must be between the two sides.
ASA and AAS
Angle-side-angle (ASA): two angles and the included side. Angle-angle-side (AAS): two angles and a non-included side. AAS follows from ASA, because if two angles match, the third must match too, since all three sum to 180 degrees.
Methods that do not work
Angle-angle-angle (AAA) does not prove congruence: a small and a large equilateral triangle have the same angles but different sizes. Side-side-angle (SSA) also fails in general, because two different triangles can share two sides and a non-included angle; this is the 'ambiguous case'.
HL for right triangles
For right triangles, hypotenuse-leg (HL) works: if the hypotenuse and one leg of one right triangle are congruent to those of another, the triangles are congruent. It is a special case where SSA does succeed, because the right angle and Pythagoras's theorem fix the third side.
CPCTC
Once triangles are proved congruent, every pair of corresponding parts is congruent, often abbreviated CPCTC: corresponding parts of congruent triangles are congruent. Proofs commonly show two triangles congruent by SSS, SAS, ASA, AAS, or HL and then use CPCTC to prove the segments or angles of interest equal.
Worked example
To measure the width of a river without crossing it, a surveyor marks points to form two triangles sharing a vertical angle at point C, with AC = DC on land. Since the vertical angles are congruent and the right angles at A and D are congruent, the triangles are congruent by ASA. By CPCTC, the inaccessible distance AB across the river equals the measurable distance DE on land.
Key takeaways
Congruent triangles match in all corresponding sides and angles, and the order of letters in a congruence statement shows the correspondence. SSS, SAS, ASA, AAS, and, for right triangles, HL prove congruence; AAA and SSA do not. The included angle or side lies between the other two parts. After proving triangles congruent, CPCTC shows any corresponding parts are congruent, which lets surveyors measure inaccessible distances.
Flashcard
Same size and shape
Check your understanding
Which condition does NOT guarantee two triangles are congruent?